1. In a large pot over medium-high heat, melt the butter. Add sliced onions and cook for about 10 minutes, stirring frequently until softened. Reduce heat to medium, add salt and sugar, and continue cooking for about 35 minutes, stirring every 5 minutes, until onions are deeply caramelized.

2. Stir in the minced garlic and cook for 20 seconds. Add the vegetable broth, vegetarian Worcestershire sauce, thyme, and bay leaf. Let the soup simmer gently over medium-low heat for 20 minutes. Remove and discard the thyme and bay leaf.

3. Stir in balsamic vinegar. Taste and season with additional salt and pepper if needed. Reduce heat to low and cover to keep warm.

4. Preheat the oven broiler, placing the rack about 6 inches from the top. Brush both sides of the bread slices with olive oil and broil for 2–3 minutes total, flipping once, until both sides are golden. Watch carefully to prevent burning.

5. Place oven-safe bowls on a baking sheet. Fill each bowl about two-thirds with hot soup. Top each with a slice of toasted bread, then evenly sprinkle with Gruyere, mozzarella, and Parmesan.

6. Broil until the cheese is melted, bubbling, and golden brown. Serve hot.